Why Positive Reals have 2 square roots
In 300 BCE the Greek mathematician Euclid of Alexandria may not have been aware proportion is preserved across direction of magnitudes. René Descartes was also unaware in 1637 when he published La Géométrie which had the the top half of this diagram copied from Euclid's Book VI Proposition 13.
Fast forward to Isaac Newton and Newton makes it known a line in an opposite direction to an 'affirmative' line may be considered 'negative'. Newton was the first to popularise the depiction of curves spanning the 4 quadrants of the Cartesian Plane. See also: http://aleph0.clarku.edu/~djoyce/elements/bookVI/propVI13.html